The reality of this trope is complicated and controversial, to put it mildly.
For starters, it would be an oversimplification to call this trope an absolute truth. Not every revolution is a Full-Circle Revolution, and there are without doubt meaningful and important differences in the motivations, methods, and goals of different political viewpoints. The fact that even opposing sides can find common ground is one of the basic reasons government of any sort is possible at all.
With that said, however, the most telling part of this trope may be that individuals favoring extreme viewpoints across the sociopolitical spectrum argue so passionately that this trope is a fallacy. Heck, moderates are not immune to this as it's often a Berserk Button for Democrats and Republicans in the United States (for example) to be told there is no true difference between the parties and are just the status quo in practice despite hostilities to each other.
Horseshoe similarities (and the gut oppositions to them) take a few forms:
PragmatismOne of the most common forms of the trope is the result of simply being pragmatic. To take the argument to the extreme, as Pragmatic Villainy and the Evil Overlord List note, it would be difficult to tell an Evil Overlord who rules the citizens of Eviltopia firmly but fairly, protects their domain from invaders, seeks to expand the prosperity and productivity of their empire, makes The Trains Run on Time, and provides Bread and Circuses to their subjects from a particularly strict Good King with a questionable passion for Spiked clothing and architecture.
Solving a similar problem (or "problem") utilizing similar tacticsSimilarly, a Necessarily Evil act by a government seeking to root out dangerous terrorists in the interest of protecting their people can be virtually indistinguishable from a Witch Hunt by Secret Police seeking to stamp out political opposition. When human nature and paranoia creep in, the difference can become erased entirely. For instance, take an official government hearing tasked with seeking out subversives, radicals, and threats to the people... and then compare this suppression of Syndicalism and Democratic Centralism under Lenin to the House Un-American Activities Committee.Likewise, later commentators find parallels between the American Red Scare and use of mass-media to develop anti-communist propaganda to be a mirror image of Stalinist persecution (albeit far less violent) and propaganda.
The complex political spectrum vs. Black and White InsanityThis is the aspect that is the Berserk Button for many with more extreme political views. Even the most open minded people used to thinking in terms of simple two-axis politics have an understandable gut reaction to a viewpoint they consider completely opposite their own being correctly identified as sharing a trait on an unfamiliar axis. The truth is that conservative and progressive, capitalist and communist, statism and individualism, and any number of others are each only individual axes of the polical spectrum. Anarcho syndicalists or libertarian socialists for instance frequently consider themselves as the opposite of fascism as they promote an immediate classless and stateless society with social freedoms for all, whereas a fascist promotes a totalitarian and unequal one. Indeed, Anarcho Syndicalists will generally defend themselves against the Horseshoe Theory by stating that it assumes the only far left tendency is Leninism which at times shared the authoritarian practices of fascism. Ironically, Anarcho Syndicalists or even Trotskyists themselves may start using the Horseshoe Theory by stating that certain Communist tendencies are traitors to the working class (an example of this would be the Trotskyist work Animal Farm comparing Stalin to the Tsar).
Unintended Consequences and Full Circle Revolutions .Many political ideologies have a tendency to unintentionally create the exact government they sought to overthrow. Liberalism at its extremes, far from being the central bulwark against authority, is in and of itself an ideology that has greatly enabled fascism and the rise of dictatorships. Liberal opposition and fear of reform and revolution is seen as a key factor in the rise of Mussolini, since the latter blackmailed the establishment into accepting him as an alternative to communism.
Liberating people from liberalismOne example of the horseshoe effect in action can be found during the Cold War.During this period both sides developed the bad habit of doing things like overthrowing communist-friendly democratic governments in favor of pro-capitalist dictators, and ousting mostly benevolent monarchies in favor of unstable nominally communist-aligned governments. Both sides claimed they were liberating the people, and the ripples of political instability reverberate to this day. During the 30s, the likes of Winston Churchill, while an anti-Nazi, nonetheless expressed vocal support for Generalissimo Franco rather than the ruling Spanish Republic. After the War, Churchill backed the pre-war Greek fascist government (some of whom became Les Collaborateurs during the Nazi Occupation) over ELAS (the Partisan movement that included communists, socialists and other liberals) who were crushed despite being La Résistance formerly supported by the Allies.
In the view of anarchists, liberals and critical anarcho-syndicalists, such as Noam Chomsky, it is possible for any ideology opposed to something else to start Jumping Off the Slippery Slope and become the very thing it opposes. Mostly by adopting features of that which it opposes. An anarchist in Ursula K. Le Guin's The Dispossessed was quoted saying "The means are the ends"—i.e., you can't use evil means to achieve good ends, since they corrupt your ends through the means which were used for achieving them. One anarchist idea has been to build a new society in the shell of the old, so by necessity it must reflect what you want to achieve. Using means which are against that would defeat the purpose. This was an anarchist criticism of Marxism which turned out to be sadly accurate.Weimar Republic for instance both the NSDAP (aka the Nazi Party) and the KPD (the communist party) ran on nationalist populism, opposed the treaty of Versailles, tried to get votes by introducing popular ballot measures and even voted and introduced measures of no confidence in unison. Of course this had to come to an end and it did when the Nazis took over in 1933, one of their first acts being to throw the communists into jail and later on communists did provide many resistance fighters, even though many communists entered the SA and the NSDAP surprisingly fast upon the Nazi takeover. A similar thing happened on the geopolitical level when both Nazi Germany and the Soviet Union were internationally isolated and looking for allies and found them in each other, before - naturally - Hitler betrayed Stalin to the surprise of precisely no one besides Stalin himself.
However, during the mid 1930s the remaining communist parties in parliamentary systems wised up and started entering tactical alliances against fascism with social democratic or even "bourgeois" parties in countries like France or Spain.
After the war, communist parties both in Western and in Eastern Europe had little in common with the extreme nationalist parties, however German fascists and classical radical leftists in the GDR tradition did have a shared hatred of Israel (the GDR only entered diplomatic relations with the Jewish state in 1989 while they had cordial relations with the PLO much earlier) which has left traces to this day in leftism and BDS (a movement calling for boycotts of Israel) has support among both the extreme left and the extreme right, due to anti-Zionism/antisemitism depending on the parties involved.
Embracing Enemy Mine and shaving off elements of Fascist and Communist ideology to form a united front or ideology leads to the development of Third Position parties as is written below.
Using the Political Compass instead of spectrumUsing the Political Compass and its quadrants of authoritarian vs. libertarian and economic left vs. right may explain why some sides can act similar. For authoritarians, regardless of their economic and social views, the point remains that using the state and collectivism are hands down the best means of promoting their end goals of a worker's paradise or a nationalist haven. For libertarians, the argument is that the government can be traced as the source of all problems because it can be used to promote a nationalist or capitalist or communist regime which gets in the way of the anarchist left and right's end goals of either a classless or free-market society.
Thus, the authoritarian right sees the libertarian right as being the enablers of the free and "decadent" order that promotes progressive social thought due to a lack of state regulating people. The libertarian right meanwhile sees the authoritarian right as being just as collectivist and interventionist as the communists they hate. Meanwhile, the authoritarian left accuses the libertarian left of just being individualists who only have personal rebellions with no end goal, while libertarian leftists accuse authoritarian leftists of being closeted reactionaries and nationalists (although some authoritarian leftists, like National Communists, are indeed nationalist and would get along with most Republicans as long as they didn't talk economics).
At the end, the compass sees the authority vs. liberty dynamic as tools and while people with opposite economic and social ideology might disagree on which system they want for humanity, they can agree on the tools used to achieve them. Therefore, the true opposites for the compass are anarcho-capitalists and communist or anarcho-syndicalists and fascists as opposed to just communists and fascists and even then someone can point out similarities between the opposite quadrants. Fascist and modern anarchist left thought are largely based around social movements instead of economic ones while anarcho-capitalist and Marxist ideology are based on economic determinism.
In any case, the compass also explains the rise of unironic Commie Nazis. Namely, while Commie Nazis are just as authoritarian as their Marxist and Fascist counterparts, they are center authoritarians due to being economically left and socially right, and even then they tend not to advocate for stateless, classless societies like Communists or ethnic supremacism like Nazis even if they skirt close (although some, like National Communists (think Romanian communist dictator Nicolae Ceaușescu), do tend to advocate for a classless society despite being socially conservative (more center-right than far-right socially, basically communist Tories) and Strasserists do advocate for Nazi-style ethnic supremacism, given that they are a branch of Nazism).